I think the key are these terminological definitions:

"In order theory, a partial order is called well-founded if the
corresponding strict order is a well-founded relation. If the order is a
total order then it is called a well-order."
https://en.wikipedia.org/wiki/Well-founded_relation

So my tree "?" in question might have no infinite descend,
but it might not belong to the same total order, as the
other sets. But how exclude "?" ? The criteria of

transitive set is not violated:
trans(A) :<=> ∀ x , y : x ∈ A ∧ y ∈ x ⇒ y ∈ A
https://de.wikipedia.org/wiki/Transitive_Menge

One can easily verify that the above is satisfied by
the set "?". So what is violated? Well this
here is violate, namely each elememt should be

transitive as well, and so on:

"hereditarily transitive sets"
h-trans(A) :<=> trans(A) & ∀x(x ∈ A => h-trans(A))

Because the third branch is not transitive:

> ? = o
> / | \
> * o o
> | |
> * o
> |
> *
>
> Or as a set:
>
> ? = {{},{{}},{{{}}}}.

So I remember Jim Burns when he posited a more
general approach, he said transfinite induction
must be satisfied.

Otherwise we can take this Quine atom x = {x},
https://math.stackexchange.com/a/2874533

And by a suitable interpretation of the circular
h-trans definition, a definition that is not well-defined
since it has no unique interpretation,

we might judge this Quine atom an ordinal.

LoL

Mild Shock schrieb:
>
> Know that one is the secret and source of all the cardinals.
> -- Abraham ibn Ezra (1153)
>
> But have mercy to me. So far I thought in ordinal
> anlysis of programs, we would simply take the
> tree of execution, and this is somehow an ordinal
>
> for a terminating program? But whats the tree
> repectively ordinal for for example 3! = 6 ?
> Are all finite ordinals the same or not?
>
> For example the ordinals 0, 1, 2, 3, with von
> Neumann succeessor are:
>
> 0 = {}
> 1 = {{}}
> 2 = {{},{{}}}
> 3 = {{},{{}},{{{},{{}}}}}.
>
> Or as trees, * = empty set, o = non-empty set:
>
> 0 =            *
>
> 1 =            o
>                |
>                *
>
> 2 =            o
>               / \
>              *   o
>                  |
>                  *
>
>
> 3 =            o
>              / |  \
>            *   o    o
>                |   / \
>                *  *   o
>                       |
>                       *
>
>
> But what about this tree, it has also no infinite decend,
> but what property is missing to make it an ordinal?
>
> ? =             o
>               / | \
>              *  o  o
>                 |  |
>                 *  o
>                    |
>                    *
>
> Or as a set:
>
> ? = {{},{{}},{{{}}}}.
>
> Why is it not an ordinal?
>
> P.S.: I tried to find an answer here, but I guess
> I am too lazy to read it. Its starts with the funny
> quote and has funny pictures in it:
>
> Trees, ordinals and termination
> N Dershowitz · 1993
> https://link.springer.com/content/pdf/10.1007/3-540-56610-4_68.pdf
>
> Mild Shock schrieb:
>> Somehow I am quite new to the ordinal analysis
>> that Alan Turing started. Nice find Gödel introduces
>> a nice notion in his incompletness paper,
>>
>> namely he says a function is primitive recursive,
>> to degree N, if the primitive recursion schema is
>> applied N times. So I guess this definition of
>>
>> factorial would then have degree 3, since 'R' is used 3 times:
>>
>> /* The R combinator */
>> natrec(_, 0, X, X) :- !.    % attention, not steadfast
>> natrec(F, N, X, Z) :- M is N-1, natrec(F, M, X, Y), call(F, Y, Z).
>>
>> plus(X, Y, Z) :- natrec(succ, X, Y, Z).
>>
>> mult(X, Y, Z) :- natrec(plus(X), Y, 0, Z).
>>
>> step((X,Y),(Z,T)) :- succ(X,Z), mult(Z,Y,T).
>>
>> factorial(X,Y) :- natrec(step,X,(0,1),(_,Y)).
>>
>> Works fine, although eats quite some computing resources,
>> i.e. 9_303_219 inferences, since it must form
>> 3268800 successors:
>>
>> ?- factorial(10,X).
>> X = 3628800.
>>
>> ?- time(factorial(10,X)).
>> % 9,303,219 inferences, 0.578 CPU in 0.617 seconds
>> (94% CPU, 16092054 Lips)
>> X = 3628800.
>>
>> Ross Finlayson schrieb:
>>> ... fourier my ass, what has it to do with ordinals ...
>>>
>>
>

On 3/1/2024 3:08 PM, Mild Shock wrote:

> So I remember Jim Burns when he posited a more
> general approach, he said transfinite induction
> must be satisfied.

In a phrase I'm a little proud of, I said
transfinite.induction is well.order in drag.
One is a simple re-write of the other.

> Otherwise we can take this Quine atom x = {x},
> https://math.stackexchange.com/a/2874533
>
> And by a suitable interpretation of the circular
> h-trans definition, a definition that is not well-defined
> since it has no unique interpretation,
>
> we might judge this Quine atom an ordinal.
>
> LoL

Ah.
But an ordinal is
a _regular_ transitive set of transitive sets.
So, not x = {x}

A regular non.empty set A holds
a disjoint element B.
∃B ∈ A: A∩B = ∅

But, if A is transitive.transitive,
each element is a subset, and
∀B ∈ A: A∩B = B
Transitive.transitive A can only be regular
if one of its elements is 0

∅ ∈ A ∧ ∅ ∈ A′ ties all the ordinals together.

It's a beautiful thing.

Lets work without regularity axiom, and
examine this naive attempt, hereditary =
my ancestors satisfied it as well:

"hereditarily transitive sets"
h-trans(A) :<=> trans(A) & ∀x(x ∈ A => h-trans(A))

Otherwise when regularity is present,
this excludes Quine atom q = {q}. When regularty is
not present, we can prove:

~h-trans(q)

Jim Burns schrieb:
> On 3/1/2024 3:08 PM, Mild Shock wrote:
>
>> So I remember Jim Burns when he posited a more
>> general approach, he said transfinite induction
>> must be satisfied.
>
> In a phrase I'm a little proud of, I said
> transfinite.induction is well.order in drag.
> One is a simple re-write of the other.
>
>> Otherwise we can take this Quine atom x = {x},
>> https://math.stackexchange.com/a/2874533
>>
>> And by a suitable interpretation of the circular
>> h-trans definition, a definition that is not well-defined
>> since it has no unique interpretation,
>>
>> we might judge this Quine atom an ordinal.
>>
>> LoL
>
> Ah.
> But an ordinal is
> a _regular_ transitive set of transitive sets.
> So, not x = {x}
>
> A regular non.empty set A holds
> a disjoint element B.
> ∃B ∈ A: A∩B = ∅
>
> But, if A is transitive.transitive,
> each element is a subset, and
> ∀B ∈ A: A∩B = B
> Transitive.transitive A can only be regular
> if one of its elements is 0
>
> ∅ ∈ A  ∧  ∅ ∈ A′ ties all the ordinals together.
>
> It's a beautiful thing.
>
>

Corr.: we cannot prove:

~h-trans(q)

See also the remark here by Andrés E. Caicedo:

Note that in the absence of foundation (= regularity),
this is a bit peculiar. For instance, if x={x}, then x
is hereditarily finite, although it does not belong to Vω.)
https://math.stackexchange.com/a/2874533

About digging into "transfinite.induction is well.order in drag"
by Jim Burns. You probably mean transfinite.induction
follows from well.order. What about the other direction?

Now my question, is assume we have no foundation,
but epsilon induction, what will happen. epsilon
induction is usually not an axiom. But what

if we stipulate it as an axiom?

Considered as an axiomatic principle, it is
called the axiom schema of set induction.
∀ x . ( ( ∀ ( y ∈ x ) . ψ ( y ) ) → ψ ( x ) ) → ∀ z . ψ ( z )
https://en.wikipedia.org/wiki/Epsilon-induction

Mild Shock schrieb:
> Lets work without regularity axiom, and
> examine this naive attempt, hereditary =
> my ancestors satisfied it as well:
>
> "hereditarily transitive sets"
> h-trans(A) :<=> trans(A) & ∀x(x ∈ A => h-trans(A))
>
> Otherwise when regularity is present,
> this excludes Quine atom q = {q}. When regularty is
> not present, we can prove:
>
> ~h-trans(q)
>
>
> Jim Burns schrieb:
>> On 3/1/2024 3:08 PM, Mild Shock wrote:
>>
>>> So I remember Jim Burns when he posited a more
>>> general approach, he said transfinite induction
>>> must be satisfied.
>>
>> In a phrase I'm a little proud of, I said
>> transfinite.induction is well.order in drag.
>> One is a simple re-write of the other.
>>
>>> Otherwise we can take this Quine atom x = {x},
>>> https://math.stackexchange.com/a/2874533
>>>
>>> And by a suitable interpretation of the circular
>>> h-trans definition, a definition that is not well-defined
>>> since it has no unique interpretation,
>>>
>>> we might judge this Quine atom an ordinal.
>>>
>>> LoL
>>
>> Ah.
>> But an ordinal is
>> a _regular_ transitive set of transitive sets.
>> So, not x = {x}
>>
>> A regular non.empty set A holds
>> a disjoint element B.
>> ∃B ∈ A: A∩B = ∅
>>
>> But, if A is transitive.transitive,
>> each element is a subset, and
>> ∀B ∈ A: A∩B = B
>> Transitive.transitive A can only be regular
>> if one of its elements is 0
>>
>> ∅ ∈ A  ∧  ∅ ∈ A′ ties all the ordinals together.
>>
>> It's a beautiful thing.
>>
>>
>

My numb nut rewriting faculty, after spying
wikipeda, takes the contrapositive (i.e replace
A->B by ~B -> ~A) of the set induction axiom:

¬ ∀ z ψ ( z ) → ¬ ∀ x ( ( ∀ y ( y ∈ x → ψ ( y ) ) → ψ ( x ) )

Now use for ψ ( z ) the formula ¬ z ∈ u, and one gets:

∃ z z ∈ u → ∃ x ¬ ( ∀ y ( y ∈ x → ¬ y ∈ u ) → ¬ x ∈ u )

Again some contrapositive:

∃ z z ∈ u → ∃ x ¬ ( x ∈ u → ¬ ∀ y ( y ∈ x → ¬ y ∈ u ))

And hence:

∃ z z ∈ u → ∃ x ( x ∈ u ∧ ∀ y ( y ∈ x → ¬ y ∈ u ))

Some last quantifier switch, and we got the regularity axiom:

∃ z z ∈ u → ∃ x ( x ∈ u ∧ ∃ y ( y ∈ x ∧ y ∈ u ))

Usually written as:

u ≠ ∅ → ∃ x (x ∈ u ∧ x ∩ u ≠ ∅)

Mild Shock schrieb:
> Corr.: we cannot prove:
>
> ~h-trans(q)
>
> See also the remark here by Andrés E. Caicedo:
>
> Note that in the absence of foundation (= regularity),
> this is a bit peculiar. For instance, if x={x}, then x
> is hereditarily finite, although it does not belong to Vω.)
> https://math.stackexchange.com/a/2874533
>
> About digging into "transfinite.induction is well.order in drag"
> by Jim Burns. You probably mean transfinite.induction
> follows from well.order. What about the other direction?
>
> Now my question, is assume we have no foundation,
> but epsilon induction, what will happen. epsilon
> induction is usually not an axiom. But what
>
> if we stipulate it as an axiom?
>
> Considered as an axiomatic principle, it is
> called the axiom schema of set induction.
> ∀ x . ( ( ∀ ( y ∈ x ) . ψ ( y ) ) → ψ ( x ) ) → ∀ z . ψ ( z )
> https://en.wikipedia.org/wiki/Epsilon-induction
>
> Mild Shock schrieb:
>> Lets work without regularity axiom, and
>> examine this naive attempt, hereditary =
>> my ancestors satisfied it as well:
>>
>> "hereditarily transitive sets"
>> h-trans(A) :<=> trans(A) & ∀x(x ∈ A => h-trans(A))
>>
>> Otherwise when regularity is present,
>> this excludes Quine atom q = {q}. When regularty is
>> not present, we can prove:
>>
>> ~h-trans(q)
>>
>>
>> Jim Burns schrieb:
>>> On 3/1/2024 3:08 PM, Mild Shock wrote:
>>>
>>>> So I remember Jim Burns when he posited a more
>>>> general approach, he said transfinite induction
>>>> must be satisfied.
>>>
>>> In a phrase I'm a little proud of, I said
>>> transfinite.induction is well.order in drag.
>>> One is a simple re-write of the other.
>>>
>>>> Otherwise we can take this Quine atom x = {x},
>>>> https://math.stackexchange.com/a/2874533
>>>>
>>>> And by a suitable interpretation of the circular
>>>> h-trans definition, a definition that is not well-defined
>>>> since it has no unique interpretation,
>>>>
>>>> we might judge this Quine atom an ordinal.
>>>>
>>>> LoL
>>>
>>> Ah.
>>> But an ordinal is
>>> a _regular_ transitive set of transitive sets.
>>> So, not x = {x}
>>>
>>> A regular non.empty set A holds
>>> a disjoint element B.
>>> ∃B ∈ A: A∩B = ∅
>>>
>>> But, if A is transitive.transitive,
>>> each element is a subset, and
>>> ∀B ∈ A: A∩B = B
>>> Transitive.transitive A can only be regular
>>> if one of its elements is 0
>>>
>>> ∅ ∈ A  ∧  ∅ ∈ A′ ties all the ordinals together.
>>>
>>> It's a beautiful thing.
>>>
>>>
>>
>

On 03/01/2024 11:38 AM, Mild Shock wrote:
> Somehow I am quite new to the ordinal analysis
> that Alan Turing started. Nice find Gödel introduces
> a nice notion in his incompletness paper,
>
> namely he says a function is primitive recursive,
> to degree N, if the primitive recursion schema is
> applied N times. So I guess this definition of
>
> factorial would then have degree 3, since 'R' is used 3 times:
>
> /* The R combinator */
> natrec(_, 0, X, X) :- !. % attention, not steadfast
> natrec(F, N, X, Z) :- M is N-1, natrec(F, M, X, Y), call(F, Y, Z).
>
> plus(X, Y, Z) :- natrec(succ, X, Y, Z).
>
> mult(X, Y, Z) :- natrec(plus(X), Y, 0, Z).
>
> step((X,Y),(Z,T)) :- succ(X,Z), mult(Z,Y,T).
>
> factorial(X,Y) :- natrec(step,X,(0,1),(_,Y)).
>
> Works fine, although eats quite some computing resources,
> i.e. 9_303_219 inferences, since it must form
> 3268800 successors:
>
> ?- factorial(10,X).
> X = 3628800.
>
> ?- time(factorial(10,X)).
> % 9,303,219 inferences, 0.578 CPU in 0.617 seconds
> (94% CPU, 16092054 Lips)
> X = 3628800.
>
> Kimono Rictus erupted while counterfeiting:
>> ... fourier my ass, what has it to do with ordinals ...
>>
>

Joseph Fourier? Fourier is famed for the Fourier-style
analysis, particularly the particular Fourier analysis.
Dirichlet and Fejer employ trigonometry.

One of it's most well-known counterparts is the
Taylor-style analysis, up after l'Hospitale's rule,
Rolle's theorem, up into Rodriguez formula.

The idea of putting them together about a
sort of Clairaut-MacLaurin for Fourier-Taylor,
about the zeros, is for that it's quite modern,
these sorts approaches.

When one mentions ordinals, as we've been
discussing ordering theory and the order-type
of ordinals and ORD the order-type of ordinals,
Cesare Burali-Forti's as it were, that the order,
the rulial, the regular, also lends to association,
the ordinance, or local laws, and the ordnance,
or, local laws.

So, when you say ORD, is it, loaded? The term?

Here it's got all the ordinals in it and contains itself.

If you didn't already know the theory of ordinals,
it's its own sort of primary element in the universe
of mathematical objects as for a theory by there being
a structure, a model theory, a model of same - ordinals,
and I suppose I've used the phrase "ubiquitous ordinals"
at least for twenty years.

It's a theory with numbers in it?

The ordering theory the axiomatic sub-field, has picked
up a lot of ground over the past decades, I don't much
know its particulars, except as with regards to that
the clock-arithmetic it makes for the modular, just like
the iota-values going zero to one, is about exactly how
it's done, here with regards usually enough to category
theory [0,1].

Some people use it instead of set theory,
it suffices for their needs.

Hey Burse, don't be counterfeiting. It's not just that
I don't like it, but governments and syndicates don't like it.
The law don't like it.

On 3/1/2024 5:11 PM, Mild Shock wrote:

> About digging into
> "transfinite.induction is well.order in drag"
> by Jim Burns. You probably mean
> transfinite.induction follows from well.order.
> What about the other direction?

Both directions.

well.order

∃ᵒʳᵈγ:p(γ) ⟹ ∃#1ᵒʳᵈβ:p(β)

∃ᵒʳᵈγ:p(γ) ⟹ ∃ᵒʳᵈβ:(p(β) ∧ ¬∃ᵒʳᵈα<β:p(α))

¬∃ᵒʳᵈβ:(p(β) ∧ ¬∃ᵒʳᵈα<β:p(α)) ⟹ ¬∃ᵒʳᵈγ:p(γ)

∀ᵒʳᵈβ:(¬p(β) ∨ ¬∀ᵒʳᵈα<β:¬p(α)) ⟹ ∀ᵒʳᵈγ:¬p(γ)

∀ᵒʳᵈβ:(​̅p(β) ∨ ¬∀ᵒʳᵈα<β:​̅p(α)) ⟹ ∀ᵒʳᵈγ:​̅p(γ)

∀ᵒʳᵈβ:(∀ᵒʳᵈα<β:​̅p(α) ⇒ ​̅p(β)) ⟹ ∀ᵒʳᵈγ:​̅p(γ)

∀ᵒʳᵈβ:(​̅p[0,β) ⇒ ​̅p(β)) ⟹ ∀ᵒʳᵈγ:​̅p(γ)

∀ᵒʳᵈβ:(​̅p[0,β) ⇒ ​̅p[0,β)∧​̅p(β)) ⟹ ∀ᵒʳᵈγ:​̅p(γ)

∀ᵒʳᵈβ:(​̅p[0,β) ⇒ ​̅p[0,β⁺¹)) ⟹ ∀ᵒʳᵈγ:​̅p(γ)

transfinite.induction

> Now my question, is assume we have no foundation,
> but epsilon induction, what will happen. epsilon
> induction is usually not an axiom. But what
> if we stipulate it as an axiom?
>
> Considered as an axiomatic principle, it is
> called the axiom schema of set induction.
> ∀x.((∀(y∈x).ψ(y))→ψ(x))→∀z.ψ(z)
> https://en.wikipedia.org/wiki/Epsilon-induction

That wiki.page assumes regular sets.
I'm not sure it does so explicitly,
but it defines ordinals as
transitive sets of transitive sets
which, without regularity, include x = {x}

On 03/01/2024 04:41 PM, Jim Burns wrote:
> On 3/1/2024 5:11 PM, Mild Shock wrote:
>
>> About digging into
>> "transfinite.induction is well.order in drag"
>> by Jim Burns. You probably mean
>> transfinite.induction follows from well.order.
>> What about the other direction?
>
> Both directions.
>
> well.order
>
> ∃ᵒʳᵈγ:p(γ) ⟹ ∃#1ᵒʳᵈβ:p(β)
>
> ∃ᵒʳᵈγ:p(γ) ⟹ ∃ᵒʳᵈβ:(p(β) ∧ ¬∃ᵒʳᵈα<β:p(α))
>
> ¬∃ᵒʳᵈβ:(p(β) ∧ ¬∃ᵒʳᵈα<β:p(α)) ⟹ ¬∃ᵒʳᵈγ:p(γ)
>
> ∀ᵒʳᵈβ:(¬p(β) ∨ ¬∀ᵒʳᵈα<β:¬p(α)) ⟹ ∀ᵒʳᵈγ:¬p(γ)
>
> ∀ᵒʳᵈβ:(​̅p(β) ∨ ¬∀ᵒʳᵈα<β:​̅p(α)) ⟹ ∀ᵒʳᵈγ:​̅p(γ)
>
> ∀ᵒʳᵈβ:(∀ᵒʳᵈα<β:​̅p(α) ⇒ ​̅p(β)) ⟹ ∀ᵒʳᵈγ:​̅p(γ)
>
> ∀ᵒʳᵈβ:(​̅p[0,β) ⇒ ​̅p(β)) ⟹ ∀ᵒʳᵈγ:​̅p(γ)
>
> ∀ᵒʳᵈβ:(​̅p[0,β) ⇒ ​̅p[0,β)∧​̅p(β)) ⟹ ∀ᵒʳᵈγ:​̅p(γ)
>
> ∀ᵒʳᵈβ:(​̅p[0,β) ⇒ ​̅p[0,β⁺¹)) ⟹ ∀ᵒʳᵈγ:​̅p(γ)
>
> transfinite.induction
>
>> Now my question, is assume we have no foundation,
>> but epsilon induction, what will happen. epsilon
>> induction is usually not an axiom. But what
>> if we stipulate it as an axiom?
>>
>> Considered as an axiomatic principle, it is
>> called the axiom schema of set induction.
>> ∀x.((∀(y∈x).ψ(y))→ψ(x))→∀z.ψ(z)
>> https://en.wikipedia.org/wiki/Epsilon-induction
>
> That wiki.page assumes regular sets.
> I'm not sure it does so explicitly,
> but it defines ordinals as
> transitive sets of transitive sets
> which, without regularity, include x = {x}
>
>
>

Like a process in time with no beginning?

The other day James Webb Space Telescope,
roundly paintcanned inflationary cosmology,
yet, even before that, the sky survey was measuring
the age of the universe every few years, and every
few years, it got hundreds of millions of years older.

Then the JWST sort of arrived at "you know it
really looks like we might have to start counting
over".

"Epoch".

A complementary notion to "Big Bang cosmology",
is, "Steady State cosmology", or, sitting next to
"Big Bang cosmology", "cyclic cosmology".

Two principles of theories of physics include
the dichotomy of unitarity and complementarity,
which is funny because one is without dichotomy
and the other is with.

2020/1/1 - 18262 = 1970/1/1

Time, then, reflects upon these foundational theories,
and anti-foundational theories, in simile, to Big Bang
theory, and Steady State theory.

Now, since scientism and logical positivism, Compte
and Boole and the Vienna circle, and Zermelo Fraenkel
and Le Maitre, with ZF set theory and Big Bang theory,
it was exactly about a century ago. 1920: a century
of hindsight, retrospect, from 2020.

So, the idea for delta-epsilonics toward zero, but
not crossing it, and least upper bound, then relying
on symmetry, sort of has the symmetry about the
origin exists for the symmetry about the origin to
exist.

Before DesCartes, one might aver the Euclid's theory,
geometry, was a bit free-er, geometry: do it anywhere
you want. Then, the notion of equipping the Euclidean
space, with a Cartesian space, makes exactly for the
notion of the ordinate itself, the ordinates and the
abscissae, which run or drop from the origin its axes,
to the curve its intercept, in these discussions of ordinals,
see arrive the notion of the ordinate, and the co-ordinates,
and that in all our real analysis, it's always based on the
co-ordinates.

So, how can there be negative numbers when first
the numbers must go all the way to infinity, if they
never reach it? The usual idea is that the comprehension
just goes ... around.

Then for DesCartes that introduces his notions of the
vortex everywhere, as anticipating particle/wave duality
with an implicit atomism, and for example: the spiral,
space-filling curve, an aspect of a: continuum.

One might imagine a faithful as possible simulacrum
of Einstein, a Zweistein: "I'm not making you say
the universe is infinite, but Space-Time is in a
continuous manifold".

So, unitarity and complementarity, the principles of
physics, sort of have the same notions in mathematics.

Ordinals: meet ordinates.

Le 01/03/2024 à 20:00, Richard Damon a écrit :
> On 3/1/24 1:29 PM, WM wrote:
>> Le 01/03/2024 à 15:44, Richard Damon a écrit :
>>> On 3/1/24 3:47 AM, WM wrote:
>>>> Le 01/03/2024 à 04:12, Richard Damon a écrit :
>>>>
>>>>> And evvery natural number is finite and thus namable and thus visible.
>>>>
>>>> That concerns potential infinity only.
>>>
>>> No, it applies to ALL Finite numbers, which of course, being finte,
>>> never actally REACH infinite, so if you want to invent a term for that
>>> as "Potential Infinity", so be it.
>>
>> But here we assume finished infinity.
>
> If we haven't finished the Infinity, you can't have you NUF(x), since it
> starts at the infinite end.

True. NUF requires completed infinity. That is the premise.

> IF you can't support "All at Once" then you can't have the Set of
> Natural Numbers to talk about, and thus we can't have your NUF.
>
> You also can show that Achilles can't pass the Tortoise, as that
> requires adding up the values "all at once" to let him catch up.

Every step in half time is enough.

Regards, WM

On 3/2/24 7:36 AM, WM wrote:
> Le 01/03/2024 à 20:00, Richard Damon a écrit :
>> On 3/1/24 1:29 PM, WM wrote:
>>> Le 01/03/2024 à 15:44, Richard Damon a écrit :
>>>> On 3/1/24 3:47 AM, WM wrote:
>>>>> Le 01/03/2024 à 04:12, Richard Damon a écrit :
>>>>>
>>>>>> And evvery natural number is finite and thus namable and thus
>>>>>> visible.
>>>>>
>>>>> That concerns potential infinity only.
>>>>
>>>> No, it applies to ALL Finite numbers, which of course, being finte,
>>>> never actally REACH infinite, so if you want to invent a term for
>>>> that as "Potential Infinity", so be it.
>>>
>>> But here we assume finished infinity.
>>
>> If we haven't finished the Infinity, you can't have you NUF(x), since
>> it starts at the infinite end.
>
> True. NUF requires completed infinity. That is the premise.

And thus must use a logic that ALLOWS for a completed infinity.

One at a time only doesn't.

>
>> IF you can't support "All at Once" then you can't have the Set of
>> Natural Numbers to talk about, and thus we can't have your NUF.
>>
>> You also can show that Achilles can't pass the Tortoise, as that
>> requires adding up the values "all at once" to let him catch up.
>
> Every step in half time is enough.

I guess you don't understand the problem.

>
> Regards, WM
>
>

Le 02/03/2024 à 15:22, Richard Damon a écrit :
> On 3/2/24 7:36 AM, WM wrote:
>> Le 01/03/2024 à 20:00, Richard Damon a écrit :

>> NUF requires completed infinity. That is the premise.
>
> And thus must use a logic that ALLOWS for a completed infinity.

Obviously there is none.
>
> One at a time only doesn't.

One at a time only is the basis of counting. Using half the time for every
step would be possible. Otherwise there is no provable bijection. Only if
each desired step (not all) can be verified, it is matematics.

Regards, WM

On 03/01/2024 08:15 PM, Ross Finlayson wrote:
> On 03/01/2024 04:41 PM, Jim Burns wrote:
>> On 3/1/2024 5:11 PM, Mild Shock wrote:
>>
>>> About digging into
>>> "transfinite.induction is well.order in drag"
>>> by Jim Burns. You probably mean
>>> transfinite.induction follows from well.order.
>>> What about the other direction?
>>
>> Both directions.
>>
>> well.order
>>
>> ∃ᵒʳᵈγ:p(γ) ⟹ ∃#1ᵒʳᵈβ:p(β)
>>
>> ∃ᵒʳᵈγ:p(γ) ⟹ ∃ᵒʳᵈβ:(p(β) ∧ ¬∃ᵒʳᵈα<β:p(α))
>>
>> ¬∃ᵒʳᵈβ:(p(β) ∧ ¬∃ᵒʳᵈα<β:p(α)) ⟹ ¬∃ᵒʳᵈγ:p(γ)
>>
>> ∀ᵒʳᵈβ:(¬p(β) ∨ ¬∀ᵒʳᵈα<β:¬p(α)) ⟹ ∀ᵒʳᵈγ:¬p(γ)
>>
>> ∀ᵒʳᵈβ:(​̅p(β) ∨ ¬∀ᵒʳᵈα<β:​̅p(α)) ⟹ ∀ᵒʳᵈγ:​̅p(γ)
>>
>> ∀ᵒʳᵈβ:(∀ᵒʳᵈα<β:​̅p(α) ⇒ ​̅p(β)) ⟹ ∀ᵒʳᵈγ:​̅p(γ)
>>
>> ∀ᵒʳᵈβ:(​̅p[0,β) ⇒ ​̅p(β)) ⟹ ∀ᵒʳᵈγ:​̅p(γ)
>>
>> ∀ᵒʳᵈβ:(​̅p[0,β) ⇒ ​̅p[0,β)∧​̅p(β)) ⟹ ∀ᵒʳᵈγ:​̅p(γ)
>>
>> ∀ᵒʳᵈβ:(​̅p[0,β) ⇒ ​̅p[0,β⁺¹)) ⟹ ∀ᵒʳᵈγ:​̅p(γ)
>>
>> transfinite.induction
>>
>>> Now my question, is assume we have no foundation,
>>> but epsilon induction, what will happen. epsilon
>>> induction is usually not an axiom. But what
>>> if we stipulate it as an axiom?
>>>
>>> Considered as an axiomatic principle, it is
>>> called the axiom schema of set induction.
>>> ∀x.((∀(y∈x).ψ(y))→ψ(x))→∀z.ψ(z)
>>> https://en.wikipedia.org/wiki/Epsilon-induction
>>
>> That wiki.page assumes regular sets.
>> I'm not sure it does so explicitly,
>> but it defines ordinals as
>> transitive sets of transitive sets
>> which, without regularity, include x = {x}
>>
>>
>>
>
>
>
>
> Like a process in time with no beginning?
>
> The other day James Webb Space Telescope,
> roundly paintcanned inflationary cosmology,
> yet, even before that, the sky survey was measuring
> the age of the universe every few years, and every
> few years, it got hundreds of millions of years older.
>
> Then the JWST sort of arrived at "you know it
> really looks like we might have to start counting
> over".
>
> "Epoch".
>
> A complementary notion to "Big Bang cosmology",
> is, "Steady State cosmology", or, sitting next to
> "Big Bang cosmology", "cyclic cosmology".
>
> Two principles of theories of physics include
> the dichotomy of unitarity and complementarity,
> which is funny because one is without dichotomy
> and the other is with.
>
> 2020/1/1 - 18262 = 1970/1/1
>
> Time, then, reflects upon these foundational theories,
> and anti-foundational theories, in simile, to Big Bang
> theory, and Steady State theory.
>
> Now, since scientism and logical positivism, Compte
> and Boole and the Vienna circle, and Zermelo Fraenkel
> and Le Maitre, with ZF set theory and Big Bang theory,
> it was exactly about a century ago. 1920: a century
> of hindsight, retrospect, from 2020.
>
> So, the idea for delta-epsilonics toward zero, but
> not crossing it, and least upper bound, then relying
> on symmetry, sort of has the symmetry about the
> origin exists for the symmetry about the origin to
> exist.
>
> Before DesCartes, one might aver the Euclid's theory,
> geometry, was a bit free-er, geometry: do it anywhere
> you want. Then, the notion of equipping the Euclidean
> space, with a Cartesian space, makes exactly for the
> notion of the ordinate itself, the ordinates and the
> abscissae, which run or drop from the origin its axes,
> to the curve its intercept, in these discussions of ordinals,
> see arrive the notion of the ordinate, and the co-ordinates,
> and that in all our real analysis, it's always based on the
> co-ordinates.
>
>
> So, how can there be negative numbers when first
> the numbers must go all the way to infinity, if they
> never reach it? The usual idea is that the comprehension
> just goes ... around.
>
> Then for DesCartes that introduces his notions of the
> vortex everywhere, as anticipating particle/wave duality
> with an implicit atomism, and for example: the spiral,
> space-filling curve, an aspect of a: continuum.
>
> One might imagine a faithful as possible simulacrum
> of Einstein, a Zweistein: "I'm not making you say
> the universe is infinite, but Space-Time is in a
> continuous manifold".
>
> So, unitarity and complementarity, the principles of
> physics, sort of have the same notions in mathematics.
>
>
>
>
>
> Ordinals: meet ordinates.
>
>

Of course ordinals are about the slenderest,
leanest, most minimal sort of sets that establish
a course-of-passage, as what it's usually called
passing through ordinals, of infinite induction,
according to their structure, their form, their
content, their model.

For example they have almost none of the
modular character of the integers, though
each is different and they're ordered, it
involves counting back-and-forth and
up-and-down and building a mememory,
a structure the content the relation embodied,
form their scaffold a model, to relate a
model of _some_ ordinals, to a model
of _some_ integers.

So, in this way building in a usual way
models of abstract algebra's finite fields,
then on up, then on out, and for each of
those all theirs, sort of results that
_eventually_ then there's a huge structure
of all their intra-relation, "integers".

You might wonder that cardinals, in
set theory, maybe they're a little fuller?
A cardinal is an equivalence class of
_all the sets in the set-theoretic universe_
that in _all the models of functions among
those in the set-theoretic universe_ that
_all those functions indicating 1-1 and onto_
that those existing from a proto-typical
set with an element after zero's: equals
cardinal "1".

So, ordinals, basically got nothing, in
structure, except next, while cardinals,
are all the structure there is that in any
way relates anything at all a set, to a
set of prototypical elements, after
cardinal "0", as it were, cardinal "1".

Then, _ordinates_, are, way, way after that,
or for analytical geometers in their theory
kind of before, that way, way after there's
Euclidean geometry, then a Cartesian basis,
an origin and axes, with a metric on those
so it's a space and norm among those so
it's an orthonormal basis, the ordinates
are arrayed on out and down all those,
also implementing ordinals.

So, ordinals, meet ordinates, ordinates, ordinals.

Jim Burns schrieb:
>> Considered as an axiomatic principle, it is
>> called the axiom schema of set induction.
>> ∀x.((∀(y∈x).ψ(y))→ψ(x))→∀z.ψ(z)
>> https://en.wikipedia.org/wiki/Epsilon-induction
>
> That wiki.page assumes regular sets.

Well I showed classicaly that the set induction
axiom schema implies the regularity axiom:

Mild Shock schrieb:
> My numb nut rewriting faculty, after spying
>
> /* set induction */
> ¬ ∀ z ψ ( z ) → ¬ ∀ x ( ( ∀ y ( y ∈ x → ψ ( y ) ) → ψ ( x ) )
>
> |
> |
> v
>
> /* set induction */
> u ≠ ∅ → ∃ x (x ∈ u ∧ x ∩ u ≠ ∅)

The only problem with this derivation, it
might not be intuitionistically valid. I might
have used a propositional or quantifier rules,

which are not accepted intuitionistically.